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An open interval is incomplete with its usual distance, yet its topology admits another metric that is complete. This guide separates completeness of one metric from complete metrisability of the underlying space.
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Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Separate metric from topology
Route 2 · Build the G-delta criterion
Route 3 · Work concrete subspaces
Route 4 · Respect the hypotheses
Route 5 · Check and continue
CHAPTER 1 OF 12 · SEPARATE METRIC FROM TOPOLOGY
1. Completeness belongs to a metric, not directly to open sets
A metric space is complete when every Cauchy sequence converges to a point of the space. A topological space is completely metrisable when at least one metric that generates its topology is complete. The distinction matters because two metrics can produce the same open sets while assigning different meanings to “Cauchy.”
Topological properties survive every compatible metric. Completeness of a named metric need not. Complete metrisability is topological because it asks whether a suitable complete compatible metric exists.
This resolves an apparent paradox: an open interval is incomplete with its usual distance, yet it is completely metrisable. The missing endpoints can be pushed infinitely far away without changing which subsets are open.
CHAPTER 2 OF 12 · SEPARATE METRIC FROM TOPOLOGY
2. The inherited metric has a simple closed-set test
Let X be a complete metric space and A a subspace with the metric inherited from X. Then A is complete under that inherited metric if and only if A is closed in X. If A is closed, any Cauchy sequence in A converges in X and its limit remains in A. If A is not closed, a point of its closure outside A is approached by a sequence from A, which is Cauchy but has no limit in A.
This theorem concerns the inherited metric. It does not say a non-closed subspace lacks every complete compatible metric. Confusing those two claims hides the role of G-delta subsets.
CHAPTER 3 OF 12 · BUILD THE G-DELTA CRITERION
3. Work the open interval with a different metric
On (0,1), the usual metric d(x,y)=|x−y| is incomplete: the sequence xn=1/n is Cauchy but converges to 0 outside the space. Now define h(x)=log(x/(1−x)), a homeomorphism from (0,1) onto the real line, and set δ(x,y)=|h(x)−h(y)|.
The metric δ generates the usual topology because h and its inverse are continuous. It is complete because any δ-Cauchy sequence maps to a Cauchy sequence in the complete real line, whose limit pulls back to (0,1). Under δ, approaching 0 or 1 requires travelling without bound.
| Feature | Usual metric d | Logit metric δ |
|---|---|---|
| Open sets | Usual interval topology | Same topology |
| Sequence 1/n | Cauchy | Not Cauchy |
| Endpoint distance | Finite approach to 0 | Distance tends to infinity |
| Complete? | No | Yes |
The example proves existence without pretending the new metric is physically natural.
CHAPTER 4 OF 12 · BUILD THE G-DELTA CRITERION
4. G-delta subsets are the exact subspaces
A G-delta set is a countable intersection of open sets. If X is completely metrisable and A is a subspace, then A is completely metrisable exactly when A is G-delta in X. When X already carries a complete metric, this theorem identifies which subspaces can acquire some complete compatible metric, not necessarily the inherited one.
Closed subsets qualify because a closed set is G-delta in a metric space: it can be written as the intersection of open distance neighbourhoods. Open subsets also qualify; an open set is itself a countable intersection by repeating it.
The theorem is stronger than the inherited-metric criterion. It includes non-closed sets such as (0,1) and the irrationals.
CHAPTER 5 OF 12 · BUILD THE G-DELTA CRITERION
5. Why the constructed metric changes near the missing boundary
Suppose A=∩n Un, where each Un is open in a complete metric space X. Points of A may approach X\Un under the ambient metric even though that boundary is excluded from A. A complete compatible metric adds penalties that grow as a point approaches those complements.
One construction combines the original distance with bounded differences of reciprocal distance-to-complement functions. A sequence that is Cauchy in the new metric cannot quietly drift toward any excluded X\Un. Its ambient limit remains inside every Un and therefore inside A.
The precise formula varies by proof, and edge cases such as an empty complement need conventions. The transferable idea is stable: record countably many ways of nearing the missing boundary and make each visible to the metric.
CHAPTER 6 OF 12 · WORK CONCRETE SUBSPACES
6. Irrational numbers give a central example
Enumerate the rationals as q1,q2,…. The irrational numbers satisfy R\Q=∩n(R\{qn}). Each R\{qn} is open, so the irrationals are a G-delta subset of the complete real line. Therefore they are completely metrisable.
Their usual inherited metric is not complete. A sequence of irrational numbers can converge to a rational. The G-delta theorem guarantees another metric with the same subspace topology that blocks this escape in its Cauchy structure.
| Space | Position inside R | Usual inherited metric | Completely metrisable? |
|---|---|---|---|
| Closed interval [0,1] | Closed, hence G-delta | Complete | Yes |
| Open interval (0,1) | Open, hence G-delta | Incomplete | Yes |
| Irrationals R\Q | G-delta | Incomplete | Yes |
| Rationals Q | Countable dense set | Incomplete | No |
This table separates the two questions that are most often merged.
CHAPTER 7 OF 12 · WORK CONCRETE SUBSPACES
7. The rationals fail by Baire category
The rational numbers with their usual topology are a countable union of singleton sets. Each singleton has empty interior and is nowhere dense in Q because Q has no isolated points. Thus Q is meagre in itself.
A nonempty completely metrisable space is a Baire space: it cannot be a countable union of nowhere-dense subsets of itself. Therefore Q is not completely metrisable. Equivalently, Q is not a G-delta subset of R.
This argument does more than observe that the usual metric on Q is incomplete. It rules out every complete metric compatible with the usual rational topology.
CHAPTER 8 OF 12 · WORK CONCRETE SUBSPACES
8. Changed ambient spaces do not change the intrinsic verdict
The statement “A is G-delta in X” appears to depend on X. For a metrisable space A embedded in a completely metrisable ambient space, however, the theorem matches an intrinsic property: complete metrisability of A. If A is completely metrisable, its image is G-delta in any suitable completely metrisable space in which it appears as a subspace.
Still, hypotheses matter. A set may be G-delta in one arbitrary ambient space and not another when the surrounding spaces lack the needed separation or completeness conditions. Always name the ambient space and the subspace topology.
Taking the metric completion of a metrisable space provides a natural test arena. The original space is completely metrisable precisely when its image is G-delta in that completion.
CHAPTER 9 OF 12 · RESPECT THE HYPOTHESES
9. Complete metrisability is not compactness
The real line is completely metrisable but not compact. The open interval is completely metrisable under the logit metric but not compact in its usual topology. Completeness plus total boundedness yields compactness for a metric, but complete metrisability alone supplies no total-boundedness claim.
Nor does complete metrisability imply local compactness: the infinite-dimensional Hilbert space ℓ² is complete and metrisable but not locally compact. Separability is another independent condition. A Polish space is both separable and completely metrisable.
Keep the adjective matched to the theorem. Replacing “completely metrisable” by “compact” or “Polish” silently adds information.
CHAPTER 10 OF 12 · RESPECT THE HYPOTHESES
10. Practical proof routes
To prove a subspace A is completely metrisable, place it in a known completely metrisable X and express it as a countable intersection of open sets. Alternatively, exhibit a complete compatible metric or a homeomorphism to a known complete metric space. Closed-subspace arguments are the quickest when available.
To disprove complete metrisability, show A is not Baire in itself, or prove it is not G-delta in an appropriate completion. Merely finding one incomplete compatible metric is insufficient; (0,1) is the standard counterexample to that faulty method.
When proposing a new metric, verify three separate facts: it is a metric, it induces exactly the original topology, and it is complete. Skipping the middle step may solve the wrong space.
Limits and common misconceptions
A convergent sequence is Cauchy for any metric, but which sequences are Cauchy can change between compatible metrics. Completeness is not preserved by an arbitrary continuous bijection; it is preserved by an isometry, while complete metrisability is preserved by homeomorphism.
The G-delta theorem requires metrisable settings of the stated kind. It should not be transferred unchanged to a non-Hausdorff topology. Sequential arguments also need care outside first-countable spaces, where sequences may not capture closure.
Finally, “there exists a complete compatible metric” does not select a unique or geometrically preferred metric. Applications may need additional properties such as boundedness, invariance or compatibility with a norm.
CHAPTER 11 OF 12 · CHECK AND CONTINUE
11. Questions readers should ask
If a subspace is not closed, must it be incomplete?
Its inherited metric is incomplete when the ambient metric is complete, but the subspace may admit another complete compatible metric.
Are all open subsets of a complete metric space completely metrisable?
Yes. They are G-delta subsets, although their inherited metric may be incomplete.
Why is Q different from the irrationals?
The irrationals are G-delta in R. The rationals are meagre in themselves and cannot carry a complete compatible metric.
Does a complete compatible metric preserve ordinary convergence?
Yes. Compatible metrics generate the same topology, so they have the same convergent sequences in a metrisable space, even though their Cauchy sequences can differ.
CHAPTER 12 OF 12 · CHECK AND CONTINUE
12. Continue through the verified topology ecosystem
Begin with the Topology root, the Gold Standard of Education and the How X Works hub. The obstruction for Q is developed through Baire Category, Nowhere-Dense Sets and Dense Open Intersections. For the role of sequences, continue to When Sequences Miss Closure and Continuity.
For nearby compactness distinctions, use Compactness and the Finite Intersection Property and Sequential Compactness and Compactness in Metric Spaces. For a warning outside Hausdorff metric intuition, read Cofinite Spaces, Compactness and Non-Unique Limits.
Authoritative technical reading: the University of Nottingham’s Metric and Topological Spaces notes define complete metrisability and prove the G-delta subspace characterisation. Jordan Bell’s notes on Polish spaces give a compatible complete-metric construction and place the theorem beside separability. Check the hypotheses before importing the result into a non-metrisable setting.
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